The generic automorphism phi of g_{6,18} as computed by calculautom6_18.red :

phi:=

mat((b(1,1),0,0,0,0,0),

    (0,b(2,2),0,0,0,0),

    (b(3,1),b(3,2),b(2,2)*b(1,1),0,0,0),

                   2
             b(3,2)                               2
    (b(4,1),----------,b(3,2)*b(1,1),b(2,2)*b(1,1) ,0,0),
             2*b(2,2)

                          2
                    b(3,2) *b(1,1)               2              3
    (b(5,1),b(5,2),----------------,b(3,2)*b(1,1) ,b(2,2)*b(1,1) ,0),
                       2*b(2,2)

    (b(6,1),b(6,2),

               2                                                  2
       - b(3,2) *b(3,1) + 2*b(4,1)*b(3,2)*b(2,2) - 2*b(5,1)*b(2,2)
     ---------------------------------------------------------------,
                                2*b(2,2)

                                                                      2
     b(1,1)*( - b(3,2)*b(3,1) + b(4,1)*b(2,2)), - b(3,1)*b(2,2)*b(1,1) ,

           2       3
     b(2,2) *b(1,1) ))


                6       10
det(phi):=b(2,2) *b(1,1)

generic derivation : delta:=

mat((xi(1,1),0,0,0,0,0),

    (0,xi(2,2),0,0,0,0),

    (xi(3,1),xi(3,2),xi(1,1) + xi(2,2),0,0,0),

    (xi(4,1),0,xi(3,2),2*xi(1,1) + xi(2,2),0,0),

    (xi(5,1),xi(5,2),0,xi(3,2),3*xi(1,1) + xi(2,2),0),

    (xi(6,1),xi(6,2), - xi(5,1),xi(4,1), - xi(3,1),3*xi(1,1) + 2*xi(2,2)))



        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  1  0  0  0  0]
adx1 := [                ]
        [0  0  1  0  0  0]
        [                ]
        [0  0  0  1  0  0]
        [                ]
        [0  0  0  0  0  0]



        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]
adx2 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  1  0]



        [0   0  0  0   0  0]
        [                  ]
        [0   0  0  0   0  0]
        [                  ]
        [0   0  0  0   0  0]
adx3 := [                  ]
        [-1  0  0  0   0  0]
        [                  ]
        [0   0  0  0   0  0]
        [                  ]
        [0   0  0  -1  0  0]



        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
adx4 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]
        [                 ]
        [0   0  1  0  0  0]



        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
adx5 := [                 ]
        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
        [                 ]
        [0  -1  0  0  0  0]


The generic nilpotent derivation : the eigenvalues are 0

xi(1,1):=0

xi(2,2):=0

by subtracting adjoints one then may suppose:

xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0,xi(6,2):=0

delta:=

[   0        0     0  0  0  0]
[                            ]
[   0        0     0  0  0  0]
[                            ]
[   0        0     0  0  0  0]
[                            ]
[   0        0     0  0  0  0]
[                            ]
[   0     xi(5,2)  0  0  0  0]
[                            ]
[xi(6,1)     0     0  0  0  0]


We denote this delta by the shortform

shortformdelta:={xi(5,2),ss,xi(6,1)}

paramindexeslist:={{5,2},{6,1}}

With the generic automorphism one gets$

shortformdeltaprimemodadg:={b(1,1)**3*xi(5,2),
ss,
b(2,2)**2*b(1,1)**2*xi(6,1)}$

deltaprimemodg(5,2):=b(1,1)**3*xi(5,2)$

deltaprimemodg(6,1):=b(2,2)**2*b(1,1)**2*xi(6,1)$

det(AUTOM):=b(2,2)**6*b(1,1)**10$

DELTAPRIMEMODADG:=

[           0                    0         0  0  0  0]
[                                                    ]
[           0                    0         0  0  0  0]
[                                                    ]
[           0                    0         0  0  0  0]
[                                                    ]
[           0                    0         0  0  0  0]
[                                                    ]
[                               3                    ]
[           0             b(1,1) *xi(5,2)  0  0  0  0]
[                                                    ]
[      2       2                                     ]
[b(2,2) *b(1,1) *xi(6,1)         0         0  0  0  0]


****************** CASE 1 : xi(5,2) NEQ 0 *************************$

Then one may suppose xi(5,2):=1 and one gets deltaprimemodg(4,2)=k by taking :$

xi(5,2):=1$

b(1,1):=k**(1/3)$

With the generic automorphism one gets$

shortformdeltaprimemodadg:={k,
ss,
(b(2,2)**2*xi(6,1)*k)/k**(1/3)}$

deltaprimemodg(5,2):=k$

deltaprimemodg(6,1):=(b(2,2)**2*xi(6,1)*k)/k**(1/3)$

det(AUTOM):=k**(1/3)*b(2,2)**6*k**3$

DELTAPRIMEMODADG:=

[         0           0  0  0  0  0]
[                                  ]
[         0           0  0  0  0  0]
[                                  ]
[         0           0  0  0  0  0]
[                                  ]
[         0           0  0  0  0  0]
[                                  ]
[         0           k  0  0  0  0]
[                                  ]
[       2                          ]
[ b(2,2) *xi(6,1)*k                ]
[-------------------  0  0  0  0  0]
[        1/3                       ]
[       k                          ]


Hence, we are reduced in the case 1 under consideration to:$

shortformdeltaprime ={1,SS,epsilon}$

where epsilon=xi(6,1) =0,1.$

****************** CASE 2 : xi(5,2) = 0 *************************$

In that case, as we suppose delta NEQ 0 one necessarily has xi(6,1) NEQ 0$

Hence we can suppose xi(6,1):=1,$

and we are reduced  to:$

shortformdeltaprime ={0,SS,1}.$